A posteriori error estimator for a mixed finite element method for Reissner-Mindlin plate

Liberman, Elsa

doi:10.1090/S0025-5718-00-01289-8

A posteriori error estimator for a mixed finite element method for Reissner-Mindlin plate
HTML articles powered by AMS MathViewer

by Elsa Liberman PDF

Math. Comp. 70 (2001), 1383-1396 Request permission

Abstract:

We present an a posteriori error estimator for a mixed finite element method for the Reissner-Mindlin plate model. The finite element method we deal with, was analyzed by Durán and Liberman in 1992 and can also be seen as a particular example of the general family analyzed by Brezzi, Fortin, and Stenberg in 1991. The estimator is based on the evaluation of the residual of the finite element solution. We show that the estimator yields local lower and global upper bounds of the error in the numerical solution in a natural norm for the problem, which includes the $H^1$ norms of the terms corresponding to the deflection and the rotation and a dual norm for the shearing force. The estimates are valid uniformly with respect to the plate thickness.

References

D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 1, 7–32 (English, with French summary). MR 813687, DOI 10.1051/m2an/1985190100071
Douglas N. Arnold and Richard S. Falk, A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal. 26 (1989), no. 6, 1276–1290. MR 1025088, DOI 10.1137/0726074
I. Babuška and A. Miller, A feedback finite element method with a posteriori error estimation. I. The finite element method and some basic properties of the a posteriori error estimator, Comput. Methods Appl. Mech. Engrg. 61 (1987), no. 1, 1–40. MR 880421, DOI 10.1016/0045-7825(87)90114-9
I. Babus̆ka and W. C. Rheinboldt, A posteriori error estimators in the finite element method, Int. J. Numer. Methods Eng. 12 (1978), 1587-1615.
R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44 (1985), no. 170, 283–301. MR 777265, DOI 10.1090/S0025-5718-1985-0777265-X
K.-J. Bathe and F. Brezzi, On the convergence of a four-node plate bending element based on Mindlin-Reissner plate theory and a mixed interpolation, The mathematics of finite elements and applications, V (Uxbridge, 1984) Academic Press, London, 1985, pp. 491–503. MR 811058
K. J. Bathe and F. Brezzi, A simplified analysis of two plate bending elements—the MITC4 and MITC9 elements, NUMETA 87 (G. N. Pande and J. Middleton, eds.), Numerical Techniques for Engineering Analysis and Design, vol. 1, Martinus Nijhoff, Dordrecht, 1987.
Franco Brezzi, Klaus-Jürgen Bathe, and Michel Fortin, Mixed-interpolated elements for Reissner-Mindlin plates, Internat. J. Numer. Methods Engrg. 28 (1989), no. 8, 1787–1801. MR 1008138, DOI 10.1002/nme.1620280806
K. J. Bathe and E. N. Dvorkin, A four-node plate bending element based on Mindlin Reissner plate theory and a mixed interpolation, J. Numer. Methods Engrg. 21 (1985),367-383.
Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
Franco Brezzi, Michel Fortin, and Rolf Stenberg, Error analysis of mixed-interpolated elements for Reissner-Mindlin plates, Math. Models Methods Appl. Sci. 1 (1991), no. 2, 125–151. MR 1115287, DOI 10.1142/S0218202591000083
Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 9 (1975), no. R-2, 77–84 (English, with Loose French summary). MR 0400739
E. Dari, R. Duran, C. Padra, and V. Vampa, A posteriori error estimators for nonconforming finite element methods, RAIRO Modél. Math. Anal. Numér. 30 (1996), no. 4, 385–400 (English, with English and French summaries). MR 1399496, DOI 10.1051/m2an/1996300403851
Ricardo Durán and Elsa Liberman, On mixed finite element methods for the Reissner-Mindlin plate model, Math. Comp. 58 (1992), no. 198, 561–573. MR 1106965, DOI 10.1090/S0025-5718-1992-1106965-0
R. G. Durán, L. Hervella-Nieto, E. Liberman, R. Rodríguez, and J. Solomin, Approximation of the vibration modes of a plate by Reissner-Mindlin equations, Math. Comp. 68 (1999), no. 228, 1447–1463. MR 1648387, DOI 10.1090/S0025-5718-99-01094-7
P. Peisker and D. Braess, Uniform convergence of mixed interpolated elements for Reissner-Mindlin plates, RAIRO Modél. Math. Anal. Numér. 26 (1992), no. 5, 557–574 (English, with English and French summaries). MR 1177387, DOI 10.1051/m2an/1992260505571
L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
R. Verfürth, A posteriori error estimators for the Stokes equations, Numer. Math. 55 (1989), no. 3, 309–325. MR 993474, DOI 10.1007/BF01390056
R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations, Math. Comp. 62 (1994), no. 206, 445–475. MR 1213837, DOI 10.1090/S0025-5718-1994-1213837-1